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on August 28, 2002
Linear algebra is typically a course taught mostly to scientists and engineers who need to use the methods of undergraduate linear algebra to calculate things. To serve that need, and to avoid abstraction, the undergraduate linear algebra course teaches matrices, more matrices, and only matrices. Only years later do students discover that they can adapt those theorems about matrices to other situations. Often they do this quite carelessly, or if they are careful, avoid doing so and resort to more difficult methods. That is because they have not read this book. If every university had a class called "Linear Algebra Done Right", fewer math students would enter graduate school thinking that linear algebra is about matrices, and fewer otherwise sophisticated physics students would deal with linear operators by saying, "Dude, pretend it's a matrix!"
Intended to follow and complement that inescapable first class in linear algebra, "Linear Algebra Done Right" emphasizes the abstract over the concrete and elegance over brute force. Eigenvalues are banished to the latter half of the book and replaced with an abstract, definition-driven development of the basic theory. Students are liberated from coordinate hell and introduced to beautiful and powerful concepts. That perspective is unnecessary and probably confusing for engineers and scientists who only need the matrix methods for calculation, but it is a natural approach for mathematicians, physicists, and others who need a deep understanding of linear algebra to support their attack on more advanced mathematics.
A note on style: Although the material is introductory, the style is slightly more sophisticated than most introductory texts. To understand the material, the student must read closely, fill in the gaps "left as an exercise", and do the problems. This is the way all advanced mathematics books must be read, and "Linear Algebra Done Right" provides a gentle introduction to that manner of reading.
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on December 4, 2001
Sheldon Axler's "Linear Algebra Done Right" is an excellent book for the strong of heart. I am an undergraduate student majoring in mathematics, and my professors are obsessed with this book. I, however, am not.
First of all, there are no solutions to the exercises at the end of each chapter, so students are left frustrated when they cannot arrive at the next step of a proof.
Second, what the author sees and deems "obvious" as far as steps and recollections are concerned is not necessarily obvious to the reader.
Axler tries to motivate readers for the proofs by offering little exercises for them to "verify." That's overkill, but to many professors and analysts, overkill in the abstract is probably necessary in order to ensure a given student's success in an advanced linear algebra course.
I'm taking one such course to fulfill my requirements for a math major, and must say these abstract/proof courses get very monotonous and, thus, ridiculously boring. The text, itself, for this book does not particularly motivate me, and I expect to consult the chapters to learn and understand concepts, not to verify info from the chaper. Essentially, the flow of the concepts is ruined by the lack of examples; how are we supposed to verify ideas when the author hasn't really even exemplified the components of them yet?
I find myself falling asleep before I even complete one or two pages in this book. The layout is dull and the propositions and theorems seem endless.
My point is the following: That which is good for the instructor is not necessarily good for the student. Students need motivation, and it is difficult to achieve this goal without offering students detailed examples, interest-catchers and solutions to the time-consuming and overwhelming exercises and concepts.
In the realm of the college curriculum, this book is average. I understand the difficulty of making abstract algebra interesting, but this notion is precisely what students need. To the student math geniuses and professors that live and breathe math, this book is a gift from the gods.
To your average math student, however, that lacks patience and the desire to give up their free time to submerge himself/herself into this book, this book, like mine, will just end up sitting on the shelf and collecting dust.
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on October 21, 2000
The text is preety terse and condense, it would be a delight if someone taking such courses from college and learned from an instructor. It would be a real bad problem if someone use the book for self study or as a reference material for more serious studies for no problems solutions were provided (this indirectly show the author definitely try to marketing the book to college instructors as teaching materials than as self-contained study materials. As a graduate student, I would stronly recommend users to consider Strang's Linear Algebra and its Application for their first choice as the real world approcah to problems and also as a stepping stone to more advanced theoretical studies in linear algebra.
For instructors this is an excellent choice for text book in the market for the price of the book is affordable to most students and meanwhile most important concepts and materials were hiding in the problem examples which will definitely challenging the most capable students and will be easy to curve the course grade in tests and exams.
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on July 3, 2003
I was very much the typical person in the target audience of this. I was a computer science major and I had a semester of linear algebra where all I learned how to solve Ax = b. Then, I happened to pick up Axler one winter evening because the title looked intriguiging. That day changed my life.
Now, I'm a pure math major and Axler is the reason. The exposition is clean and very elegant. By minimizing the use of matrices in his proofs, he presents the subject of linear algebra as an elegant piece of mathematics rather than a subject "spoilt" by applications. He starts with a study of vector spaces and then moves onto transformations, eigenvalues, inner product spaces, etc. all the way upto the jordan form. All along, the use of matrices in minimal. In fact, he introduces them quite late in the book just as a convenient notation and nothing else. This is an admirable aspect because it simplies a lot of the proofs. The proof that every linear operator over a finite-dimensional vector space has an eigenvalue is breathtakingly short and simple. He uses determinants in the last chapter of the book and there too, does an excellent job. (although the point of writing this book was NOT to use determinants, his exposition about determinants is itself one of the best ones I've seen).
Get this book if you wish to understand the theory. It's a typical higher level math text - definition, theorem, proof, exercises (most of which are theorems). If you like math, you won't regret this.
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on October 13, 2001
This is an example of an extremely elegant book that does lose sight of the real world (Euclidean Spaces). For an undergraduate math major is one of the best books that I have ever seen. I really would love to teach a course from it. The book does not meet all needs. This is the beauty of the book, it written for people wishing to learn the basics of Linear Algebra from a mathematical point of view and it does it wonderfully. Most books, these days, try to do so much that they end up doing nothing. This book is not suitable for a first course in Linear Algebra. It rather important that the reader have some prior knowlege in order to fully appreciate some of the abstractions. Many second Linear Algebra courses try to concentrate on applications. This book is not good for this purpose. The book does provide a wonderful foundation to build the applications later. I am in complete agreement with the author's contention that it better to look at Linear Transformations than matrices. However, my teaching experience tells that before one does this one should look at the properties of matrices first and then translate the properties in terms of Linear Transformations.
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on January 31, 2000
I've seen many linear algebra books and this is by far the best treatment of them all. After going through this book one wonder why most linear algebra presentations don't follow Axler's sound and more reasonable approach. It leaves Hoffman & Kunze in the dust (although you may still want to hang on to Hoffman since it contains some material not found in Axler).
Not only is Axler's approach sound, but his writing is very lucid and clear as well. You will never leave a proof feeling unsatisfied or confused; it almost reads like a book. I wish all math books were written this way.
My only gripe with the book is the lack of solutions to the problems. Those who use the book for self-study will feel particular frustrated in this regard. I hope some effort is taken to assuage this problem in future editions. Also, more material on linear functionals and multilinear mappings (tensors) would be nice.
In summary, this is an outstanding book; I highly recommend it.
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on June 2, 2004
This is a short, elegant presentation of linear algebra appropriate for upper level undergraduate math majors with a theoretical bent. The student has perhaps taken a linear algebra course designed for engineers and scientists. Such a student is comfortable reading mathematics and writing proofs. It is meant to be read and re-read until the ideas are absorbed. The exercises are relatively easy and no answers are provided. With exercises of this sort you generally know if you are on the right track and they require you to understand the presentation in the text and process the ideas in a straight forward way.
Of course, there is nothing in this book about applications or the computational aspects of applying linear algebra.
The price is right. This could be a very useful purchase even if it's not assigned as a text.
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on March 27, 2002
I've only loooked through this book a bit, but I found the proofs to be very enlightening. It presents the ``correct'' view of linear algebra a the study of vectors spaces, not the study of R^n, and n x m matrices. The book introduces matrices towards the end for a very good reason: matrices aren't that important. The real substance of linear algebra: linear operators and vector spaces. Introducing linear operators as matrices would be like defining a homomorphism on a group by giving what the homomorphism does to the presentation for the group. An idiotic and counterintuitive method of defining homomorphisms. Yet in combinatorial group theory, it is helpful sometimes to do this. Much as it is sometimes helpful to work with matrices--but certainly not from the start.
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on January 31, 2004
This book is very well written and a pleasure to read. I used this book for my second linear algebra course. It was a wonderful account on finite-dimensional vectorspaces and finite-dimensional operators. Axler's approach of not using determinants is most efficient and very helpful to "see what's going on." This book isn't meant for those that want an applied course in linear algebra; only abstract material here! It's meant for junior/senior math majors who have some "math maturity." While Axler's very careful with the presentation, those with little experience with reading and writing proofs may find it challenging. If you're into math, pick it up and have fun!!!
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on December 5, 2000
Axler's questions at the end of each chapter reads like a book of Zen Koans, each question is short, elegant, and extrodinarily thought-provoking. Those who dismiss the book have NOT done the problems and will thus not understand this man's inherent zen-like understanding of linear algebra. In addition, take the Axler Test for yourself: Look at the way any topic in linear algebra is treated in any other book, then see the way Axler treats it. I assure you, you will be complety blown away by his clarity, elegance, and brevity. This book is for everyone, even non-math majors who wish to see an example of extreme clarity of thought in action.
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